8. Find the compounded ratio of the following:

Class 10 | Frank | ICSE | Maths | Ratio and Proportion

The ratio is used for comparing two quantities of the same kind.

The ratio formula for two numbers says a and b is given by a:b or a/b. When two or more such ratios are equal, they are said to be proportion.

The concept of ratio and proportion is majorly based on ratios and fractions.

(i) $15:16$ and $8:5$

Solution:-

Given ratio can be written as,

${15}/{16}\;$ and ${8}/{5}\;$

$={15}/{16}\;\times {8}/{5}\;$

$=\left( 15\times 8 \right)/\left( 16\times 5 \right)$

$={\left( 3\times 1 \right)}/{\left( 2\times 1 \right)}\;$

${3}/{2}\;$

Therefore, the compounded ratio of $15:16$ and $8:5$ is $3:2$ .

(ii) $\left( {{a}^{2}}-{{b}^{2}} \right):\left( {{a}^{2}}+{{b}^{2}} \right)$ and $\left( {{a}^{4}}-{{b}^{4}} \right):{{\left( a+b \right)}^{4}}$

Solution:-

Given ratio can be written as,

${\left( {{a}^{2}}-{{b}^{2}} \right)}/{\left( {{a}^{2}}+{{b}^{2}} \right)}\;$ and ${{{\left( {{a}^{4}}-{{b}^{4}} \right)}/{\left( a+b \right)}\;}^{4}}$

$={\left( {{a}^{2}}-{{b}^{2}} \right)}/{\left( {{a}^{2}}+b{}^{2} \right)}\;\times {{{\left( {{a}^{4}}-{{b}^{4}} \right)}/{\left( a+b \right)}\;}^{4}}$

We know that, $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a+b \right)\left( a-b \right)$

$=\left( \left( a+b \right)\left( a-b \right) \right)/\left( {{a}^{2}}+{{b}^{2}} \right)\times \left( \left( {{a}^{2}}+{{b}^{2}} \right)\left( {{a}^{2}}-{{b}^{2}} \right) \right)$

$={\left( \left( a-b \right)\left( a+b \right)\left( a-b \right)\left( a+b \right) \right)}/{\left( {{\left( a+b \right)}^{2}}{{\left( a+b \right)}^{2}} \right)}\;$